Is this an SARIMA(0,0,0)x(0,1,4)_12? I found someone's quick-and-dirty forecast of a variable $x$:
$$\hat{x}_t = x_{t-12} + \frac{1}{4}\Delta_{12}\left(x_{t-1} + x_{t-2} + x_{t-3} + x_{t-4} \right)$$ 
Can this be viewed as an "optimal" forecast from some SARIMA model? I see that this is very close to a SARIMA$(0,0,0)$x$(0,1,4)_{12}$
Framing my question differently, is the following model in the SARIMA family?
$$\Delta_{12}x_t = \frac{1}{4}\Delta_{12}\left(x_{t-1} + x_{t-2} + x_{t-3} + x_{t-4} \right) + \epsilon_t$$ 
 A: You're headed in the right direction. The example you give is an $SARIMA(4,0,0)\times(0, 1, 0)_{12}$ model.
To see this note that the difference operator is a linear operator and that you can algebraically manipulate it as if it were a real number. Thus, 
$$\begin{align} & \Delta_{12}x_{t}  = \frac{1}{4}\Delta_{12}(x_{t-1} + x_{t-2} + x_{t-3} + x_{t-4}) + \epsilon_{t} \\
 \implies & \Delta_{12}x_{t} - \frac{1}{4}\Delta_{12}(x_{t-1} + x_{t-2} + x_{t-3} + x_{t-4}) = \epsilon_{t} \\
\implies & \Delta_{12}\left(x_{t} - \frac{1}{4}x_{t-1} - \frac{1}{4}x_{t-2} - \frac{1}{4}x_{t-3} - \frac{1}{4}x_{t-4}\right) = \epsilon_{t}
\end{align}$$
In other words, the first seasonal difference of the process is an $AR\left(4\right)$ model.
For an illustration of this model, see the following R code. Notice that 0.25 is within one standard error of each $AR$ coefficient. You may also notice that the $AIC$ and related values are much lower than other attempted model fits.
library(forecast)
set.seed(30)
# Select the last 500 elements so as to allow a "warm-up period".
x <- w <- ts(rnorm(1000)[501:1000], freq = 12)
for (t in 17:500) {
  x[t] <- x[t-12] + 0.25*((x[t-1] - x[t-13]) + (x[t-2] - x[t-14]) + 
                            (x[t-3] - x[t-15]) +(x[t-4] - x[t-16])) + w[t]
}
model <- Arima(x, order = c(4, 0, 0), seasonal = c(0, 1, 0))
fcst <- forecast(model)
#print(model)
#print(fcst)

